Deformation theory of abelian categories

@article{Lowen2004DeformationTO,
  title={Deformation theory of abelian categories},
  author={Wenty T. Lowen and Michel van den Bergh},
  journal={Transactions of the American Mathematical Society},
  year={2004},
  volume={358},
  pages={5441-5483}
}
  • W. T. Lowen, M. Bergh
  • Published 12 May 2004
  • Mathematics
  • Transactions of the American Mathematical Society
In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems. 
View PDF on arXiv
102 Citations
Highly Influential Citations
6
Background Citations
32
Methods Citations
9
Results Citations
3

102 Citations

Obstruction Theory for Objects in Abelian and Derived Categories
ABSTRACT In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular…
The bar derived category of a curved dg algebra
Twisted Deformation Quantization of Algebraic Varieties (Survey)
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf OX. These are stack-like…
Twisted deformation quantization of algebraic varieties
Deformations of categories of coherent sheaves via quivers with relations
We give an explicit combinatorial description of the deformation theory of the Abelian category of (quasi)coherent sheaves on any separated Noetherian scheme X via the deformation theory of path…
Hochschild cohomology of abelian categories and ringed spaces
Deformations and Fourier-Mukai transforms
The aim of this paper is twofold: First we give an explicit construction of the infinitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly we…
t-structures and twisted complexes on derived injectives
Deformations of algebroid stacks
Compact moduli of noncommutative projective planes
We introduce the notion of the moduli stack of relations of a quiver. When the quiver with relations is derived-equivalent to an algebraic variety, the corresponding compact moduli scheme can be…
...
...

References

SHOWING 1-10 OF 35 REFERENCES
Obstruction Theory for Objects in Abelian and Derived Categories
ABSTRACT In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular…
Hochschild cohomology of abelian categories and ringed spaces
Abstract Hilbert Schemes
In analogy with classical projective algebraic geometry, Hilbert functors can be defined for objects in any Abelian category. We study the moduli problem for such objects. Using Grothendieck's…
The Spectrum of a Module Category
Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite…
A generalization of the Gabriel–Popescu theorem
On the Deformation of Rings and Algebras
CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5.…
Algebraic Cohomology and Deformation Theory
We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, in…
The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set
To each presheaf (over a poset) of associative algebras A we associate an algebra A!. We define a full exact embedding of the category of (presheaf) A-bimodules in that of A!-bimodules. We show that…
A Brown representability theorem via coherent functors
Handbook of Categorical Algebra
The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of…
...
...